Relationships (Direct, Linear, Etc) Graphs, Best-Fit Lines
Relationships between two variables can be classified into two major categories: direct and inverse. A direct relationship is a positive relationship between two variables in which they both increase or decrease in conjunction. An example of direct relationship would the relationship between the number of people attended the party and the food consumption at a party. At the party, the more people there are (increase of independent variable x) will lead to more food consumption (increase of dependent variable y.) On the other hand, an inverse relationship is a negative relationship between two variables in which one decreases when the other increases. An example of an inverse relationship would be the relationship between the number of workers and the time it will take the workers to finish one job. The more workers that are hired, the faster the job should be done; otherwise why bother hiring so many? So in this case, as one variable (number of workers) increases, the other variable (hours the workers take to finish the job) will decrease. Relationships are not only used in math, they are also very useful in physics as well. __TOC__ #''Linear Relationship'' A linear relationship is a type of direct relationship. Linear equations can be written in the form of y=mx+b, in which x is the independent variable, y is the dependent variable, m is the slope, and b as the y-intercept. In a linear relationship, there is always a constant multiplying one of the variables to change the other variable in a proportion. Back to the party food consumption example aforementioned, if every person consumes 2 plates of food, then the final number of plates will be resulted by multiplying the number the people by 2; 2 being the constant in this situation. Linear equations appear to be straight lines in a xy-coordinate graph. When the constant is positive, the line will extend towards the directions of northeast and southwest. When the constant is negative, the line will extend towards northwest and southeast. Linear Relationships in Physics In physics, many variables are in linear relationship. Take the equation for velocity, v=d/t, for example, there is a direct relationship between velocity and distance. Time in this case, serves as the constant: every distance has to be multiplied with 1/t to result to the velocity. When distance increases or decreases, the resulting velocity must increase or decrease in conjunction. Quadratic Relationship Quadratic relationships describe the relationship of two variables vary, directly or inversely, while one of the variables are squared. The word quadratic describes something of or relating to the second power. When it is a directly relationship will result to the shape of half of a parabola. In such a case, the two variables vary directly because they increase/decrease in conjunction. But they are described differently from a linear relationship because process of raising one of the variables to the second degree changes the rate of change every time. Quadratic Relationships in Physics Not all formulas in physics are linear; there are many that are quadratic as well. Just to name a few: * Centripetal Acceleration: ac = v2/r * Kinetic Energy: KE = ½ mv2 *Law of Universal Gravitation: Let’s explore an inverse quadratic relationship with the Law of Universal Gravitation. In the equation, d is the distance between the centers of the masses and G is the universal constant—one that is the same everywhere. According to Newton’s equation, if the mass of a planet near the sun were doubled, the force of attraction would be doubled. Similarly, if the planet were near a star having twice the mass of the sun, the force between the two bodies would be twice as great. In addition, if the planet were distance from the sun, the gravitational force would be only one quarter stronger. In this law, the distance varies inversely quadratic to the gravitational force and varies directly quadratic to G and the two masses. Line of Best Fit In reality, not every piece of data collected will fit into the formula perfectly. But if the experiment really works to the idea of the theorem, then a general shape of the graph of the equation will show up. In this case, a line of best fit should be drawn to finalize the data collection. A line of best fit is a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points. The use of line of best fit allows one to see the trend of the graph and from that, one will be able to continue the trend for prediction. How to Draw a Line of Best Fit By Hand # Prepare a scatter plot of the data. # Using a ruler, position the ruler so that the plotted points are as close to the strand as possible. # Find two points that you think will be on the "best-fit" line. # Calculate the slope of the line through your two points. # Write the equation of the line. How to Draw a Line of Best Fit By Calculator Regents Questions Relating to Relationships Regents style questions relating to relationships will not be as direct as asking “What is the relationship between momentum and velocity in the equation p=mv?” Instead, the question is most likely to show the graph of the relationship. For the example with p=mv, the correct answer will be choosing the positive linear graph. Sample Questions (Taken from Physics Regents January 2004) The correct answer is (3). The equation used here is . Variable L and R are directly related. References #“Linear Relationship of Velocity and Distance.” 'The Physics Classroom''. Tom Henderson. 12 Jun 2006 .'' #“Force vs. Location.” Zitzewits, Paul W. Glencoe PHYSICS Principles and Problems (1999). #“Line of Best Fit.” Oswego City School District Regents Exam Prep Center. #“Physics Regents June 2005” Oswego City School District Regents Exam Prep Center. External links *Sparknotes: SAT Physics - Practice Questions 1. Enter the data in the calculator lists. Place the data in L1 and L2. STAT, #1Edit, type values into the lists. 2. Prepare a scatter plot of the data. Set up for the scatterplot. 2nd StatPlot - choices shown at right. Choose ZOOM #9 ZoomStat. Graph shown below. 3. Have the calculator determine the line of best fit. STAT → CALC #4 LinReg(ax+b) Include the parameters L1, L2, Y1. (Y1 comes from VARS → YVARS, #Function, Y1) You now have the values of a and b needed to write the equation of the line of best fit. 4. Graph the line of best fit. Simply hit GRAPH. To get a predicted value within the window, hit TRACE, up arrow, and type the desired value.